Method and system for predicting process information with a parameterized model

A method for determining phases that satisfy sets of constraints for measured intensity of multi-wavelength information, the method comprising:

• obtaining measured intensity data associated with a substrate, and wavelength information associated with the measured intensity data;
• determining, based on dimensional data in a latent space of a parameterized model, a latent space of intensity data;
• constraining the latent space with the wavelength information associated with the measured intensity data;
• determining, based on the constrained latent space, an optimum latent space that satisfies constraints related to the wavelength information;
• and
• predicting, with the parameterized model using the optimum latent space, phase data for the measured intensity data.

A method for predicting process parameter data using a parameterized model, the method comprising:

• determining, based on dimensional data in a latent space of the parameterized model, a latent space of a given input;
• obtaining prior information associated with the given input;
• constraining the latent space with the prior information associated with the given input to limit a search of an optimum latent space for the given input;
• applying an optimization algorithm to the constrained latent space to determine the optimum latent space that satisfies the prior information related constraints;
• and
• predicting, by transforming the optimum latent space via the parameterized model, process parameter data associated with the given input.

A method for predicting process parameter data using a parameterized model, the method comprising:

• determining, based on dimensional data in a latent space of the parameterized model, a latent space of a given input;
• obtaining prior information associated with the given input;
• constraining the latent space with the prior information associated with the given input to limit a search of an optimum latent space for the given input;
• applying an optimization algorithm to the constrained latent space to determine the optimum latent space that satisfies the prior information related constraints;
• and
• predicting, by transforming the optimum latent space via the parameterized model, process parameter data associated with the given input.

A method for training a parametrized model configured to predict process parameter for a given input, the method comprising:

• obtaining a training data set comprising a first image representing a first parameter data and a second image representing a second parameter data,
• encoding the first image and the second image in the training set into dimensional data in the latent space,
• supplementing the latent space with prior information associated with the first image and the second image;
• and
• transforming the dimensional data in the supplemented latent space into recovered versions of images corresponding to the training set, and
• training the parametrized model by adjusting model parameters based on comparison between the recovered version of the images, and the first and the second images.